Fluctuation relations for irreversible emergence of information

Information theory and Thermodynamics have developed closer in the last years, with a growing application palette in which the formal equivalence between the Shannon and Gibbs entropies is exploited. The main barrier to connect both disciplines is the fact that information does not imply a dynamics, whereas thermodynamic systems unfold with time, often away from equilibrium. Here, we analyze chain-like systems comprising linear sequences of physical objects carrying symbolic meaning. We show that, after defining a reading direction, both reversible and irreversible informations emerge naturally from the principle of microscopic reversibility in the evolution of the chains driven by a protocol. We find fluctuation equalities that relate entropy, the relevant concept in communication, and energy, the thermodynamically significant quantity, examined along sequences whose content evolves under writing and revision protocols. Our results are applicable to nanoscale chains, where information transfer is subject to thermal noise, and extendable to virtually any communication system.

the so-called sequencedependent partition function [3]: As explained earlier [2,3], the standard partition function, which defines the equilibrium probabilities p ν = exp (−βE ν ) /Z, does not restrict how to access each final configuration. This formalism comprises the concept of revision, which includes the proofreading and the editing of the information in the chain [2,4]. Ensembleaverage equilibrium thermodynamic functions are describable in a similar fashion as the directional ones, this time by taking expected values with distribution p ν , namely,

Proofs of Fluctuation relations
We consider a chain-like system that is initially configured as a microstate ν m in equilibrium; that is, ν m belongs to an ensemble of statistically similar microstates in the absence of a protocol. Then, the chain evolves to a final microstate ν n under writing, λ = D, see Fig. 1 in the main text. As for the backwards evolution (resetting), the chain transforms from ν n in equilibrium to ν m , following the inverse protocol, D −1 .

Chain with constrained energy changes
The system is prepared in isolation, with a defined energy both for the forward and reverse evolutions at temperature T . It then grows into a (micro)state at the same temperature by exchanging a precise energy. The writing protocol transforms the chain from an ensemble of M ≤ |X | m sequences with expected entropy S(m) into a microstate with entropy S (D) ν (n). In the reverse evolution (resetting), the chain starts in an equilibrium defined by N ≤ |X | n available sequences and entropy S(n), and transforms into a microstate with entropy S (D −1 ) ν (m). We now demonstrate the first theorem in the main text, Eq. (1). The probability of the initial state in terms of the microcanonical entropy is p ν (m) = 1/M = exp (−S(m)/k) and the directional probability for a single trajectory p [2,5,6]. The change in entropy for a chain constructed under writing is ∆S where S ν (m) = S(m) in the initial conditions described above. Then: which yields: We next demonstrate Eq. (2) in the main text. The transition forward probability, p F (m n), equals the probability exp (−S(m)/k) of starting in equilibrium times the probability exp −S (D) ν (n)/k of ending in a microstate through irreversible writing. The transition reverse probability follows the same rationale, namely, Then, the ratio of forward and reverse probabilities read ν (m), given that we start at the final microstate and recover the initial one by removing objects (or by resetting them to their initial values) in sequence ν n−m . Using ∆S = S(n) − S(m) and ∆S ν (m), we find: which can be expressed as: It is straightforward to see that Eq. (S4) (i.e. Eq. (1) in the main text) can also be obtained from Eq. (S7) (Eq. (2) in the main text). Certainly, by summing over all sequences on both sides of Eq. (S7), we find: Chain in contact with a thermal reservoir The system starts in equilibrium with a thermal bath at temperature T , with which it can exchange energy at all times. The Helmholtz free energy characterizes the equilibrium initial states in these conditions, both for the forward and the reverse evolutions. The change in free energy for a chain constructed under writing is ∆F , independent of the sequence [2,5]. Then: where ∆E ν (m, n) = E ν (n) − E ν (m) is the energy change between the initial and final sequences and ∆U (m, n) = U (m) − U (n) the corresponding expected energy change in equilibrium conditions. Equation (S10) closes the proof to Eq. (4) in the main text. Next, we derive Eq. (5) in the main text. The transition forward canonical probability, P F (m n), equals the probability p ν (m) of starting in equilibrium times the probability p (D) ν (n) of ending in a general microstate. The transition reverse canonical probability follows the same argument, namely, P R (m n) = p ν (n) × p (D −1 ) ν (m). Then, the ratio of forward and reverse probabilities reads: . (S11) The partition functions in Eq. (S11) can be expressed in terms of the free energies, where F ν (m), given that we start at the final microstate and recover the initial one by removing objects (or by resetting them to their initial values) in sequence ν n−m . Then where we have used that ∆F (m, n) = F (n) − F (m). Within the canonical formalism, this result can also be expressed in terms of the entropy [2], more significant in information theory: It is straightforward to see that Eq. (4) in the main text can also be obtained from Eq. (5) in the main text, it is sufficient to show herein that Eq. (S9) emerges from Eq. (S16). Certainly, by summing over all sequences on both sides of Eq. (S16), we find: (S18)

Expected Work and Heat
The expression for the protocol-dependent ensemble-average work (Eq. (14) in the main text) is: where we have used that the equilibrium Helmholtz free energy is independent of the sequence, as shown above. The corresponding equilibrium work (see the main text) follows the same reasoning by adding that F ν (n) = −kT ln Z(n) = F (n). Then W (m Concerning the expressión for the protocol-dependent heat (Eq. (11) in the main text), since the system is initially prepared in isolation, the entropy just depends on the number of available configurations, not on the precise sequence, S ν (m) = S(m). Then (S20) appears similarly as W (λ) (m n) by replacing F for T S, with T a constant. The corresponding equilibrium heat is immediate for initial and final isolated conditions, Q(m n) = T (S(n) − S(m)).

Gallavotti-Cohen theorem for information chains
We introduce the rate, σ (λ) , at which the chain exchanges heat with the bath under a certain protocol λ from the dissipated heat Q where ∆t is a time interval. With this definition, it is straightforward to derive the Gallavotti-Cohen expression [7], emerging from the so-called Fluctuation Theorem [8,9], for an information chain that is physical. Gallavotti-Cohen equality connects the forward and reverse heat-rate probability distributions for steady-state, non-equilibrium processes. It appears naturally within our formalism by following a similar argument as the one above for Eq. (